Discrete Mathematics

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This will cover the following topics

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Module 1: Logic and Proofs

Logic is the basis of all mathematical reasoning, and of all automated reasoning. It has practical applications to the design of computing machines, to the specification of systems, to artificial intelligence, to computer programming, to programming languages, and to other areas of computer science, as well as to many other fields of study.
Learning Objectives

By the end of this module, you should be able to:

Required Readings

Rosen, Kenneth H., Discrete Mathematics and its Applications, 8th Edition ISBN: 978-1-259-67651-2

Chapter 1

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Module 2: Sets, Functions and Rates of Growth

Many important discrete structures are built using sets, which are collections of objects. Among the discrete structures built from sets are combinations, unordered collections of objects used extensively in counting; relations, sets of ordered pairs that represent relationships between objects, and finite state machines, used to model computing machines. Functions also play important roles throughout discrete mathematics. They are used to represent the computational complexity of algorithms, to study the size of sets, to count objects, and in a myriad of other ways.

By the end of this module, you should be able to:

References

Rosen, Kenneth H., Discrete Mathematics and its Applications, 8th Edition ISBN: 978-1-259-67651-2

Chapters 2 (2.1 -2.3) and 3 ( 3.2)

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Module 3: Number Theory

The part of mathematics devoted to the study of the set of integers and their properties is known as number theory. In this module we will develop some of the important concepts of number theory including many of those used in computer science.
Learning Objectives

By the end of this module, you should be able to:

References

Rosen, Kenneth H., Discrete Mathematics and its Applications, 8th Edition ISBN: 978-1-259-67651-2

Chapters 4.1 - 4.4

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Module 4: Sequences and Series, Mathematical Induction, Basic Combinatorics and Probability

This module introduces the notion of a sequence, which represents ordered lists of elements. We will introduce some important types of sequences, and we will show how to define the terms of a sequence using earlier terms. We will also introduce the notation used to express summations, which represent the addition of terms from a sequence. The relative sizes of infinite sets can be studied by introducing the notion of the size, or cardinality, of a set: we say that a set is countable when it is finite or has the same size as the set of positive integers. We will study the concept of mathematical induction and apply proofs using induction.

Suppose that a password on a computer system consists of six, seven, or eight characters. Each of these characters must be a digit or a letter of the alphabet. Each password must contain at least one digit. How many such passwords are there? The techniques needed to answer this question and a wide variety of other counting problems will be introduced in this section. We will also solve linear recurrence relations and consider the basics of discrete probability.

By the end of this module, you should be able to:

References

Rosen, Kenneth H., Discrete Mathematics and its Applications, 8th Edition ISBN: 978-1-259-67651-2

Chapters 2 (2.4, 2.5) and 5.1, 5.2, 6.1 - 6.5, 7.1, 7.2, 8.1, 8.2